Cyclotron

A cyclotron is a kind of circular particle accelerator. In a cyclotron, a charged particle is accelerated along a spiral path under the action of a static magnetic field and an alternating electric field. The charged particle is inserted in a cyclotron such that its direction of motion is perpendicular to the static magnetic field. The magnetic field causes the particle to make rotations and the electric field accelerates the particle after each rotation. The electric field is generated by a high-frequency alternating voltage. The frequency of the alternating voltage is matched with the cyclotron frequency for the charged particle and generally, it is kept constant. 

Frequency of Rotation of a Charged Particle in a Uniform Magnetic Field

The theory of cyclotron is based on the interaction of a charged particle with electric and magnetic fields. The magnetic force on a particle of charge q, moving with velocity v due to a uniform magnetic field B is given by,

F = q v x B

When a charged particle moves perpendicular to a constant magnetic field with speed v, the magnitude of the magnetic force is,

F = q

νν

B sin 90°

F = q

νν

B

This force acts in a direction perpendicular to both the velocity of the particle and the magnetic field.

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The Circular Path of a Charged Particle

The particle starts to rotate in a circular path of radius r such that the magnetic force serves as the centripetal force of that circular path. The centripetal force

FcFc has magnitude,

Fc= mν2rmν2r

Here, m is the mass of the particle and r is the radius of the circular path. The centripetal force is equal to the magnetic force i.e.

FcFc= Fmν2rmν2r

= qννBνν

= qBrmqBrm

The angular velocity (cyclotron angular frequency) is given by,

ωω= νrνr

ωω= qBmqBm

The frequency of the rotation namely cyclotron frequency is,

f = ω2πω2π

f = qB2πmqB2πm

This is the cyclotron frequency formula.

Operating Principle of Cyclotron

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• In a cyclotron, two hollow “D” shaped electrodes are placed face to face with a small gap, inside a vacuum chamber. An alternating voltage is applied between the “dees” across the gap. A uniform magnetic field is applied perpendicular to the plane of the electrodes. The “dees” have a cylindrical space for the particles to move.

• Charged particles are injected at the center of the cylindrical space (shown by a dot in the figure). If the particles would have a constant velocity, they would rotate in circles of constant radii. But an alternating voltage of high frequency is applied across the gap. The frequency is set such that the charged particles make a semicircle during a single cycle of the alternating voltage. In other words, the frequency of the ac voltage must match with the cyclotron frequency of the particles given by the cyclotron formula,

f = qB2πmqB2πm

Here, q is the charge, is the mass of the particle and B is the magnetic field strength.

• Each time a particle completes a semicircle inside a dee and approaches the other dee, the polarity of the voltage flips. The particle gets accelerated towards the other dee due to the electric field created by the ac voltage and its velocity increases.

• Since the frequency remains constant, the particle starts to move in a circle of a larger radius. The particle’s trajectory takes the shape of a spiral of increasing radius. With each full cycle, the radius increases, and the velocity also increases. This process continues until the radius of the trajectory approaches the radius of the cylinder and the accelerated particles are passed through an exit at the end of the cylinder. The radius of the cylinder must be set such that the desired velocity of the particles can be reached.

Applications of Cyclotron

Cyclotrons are much more effective than linear accelerators because cyclotrons accelerate the particles several times in a single set up and due to their cylindrical shape, less space is required as compared to linear accelerators. Some of the uses of cyclotron are listed below,

• Cyclotrons are widely used to accelerate charged particles in nuclear physics experiments and use them to bombard atomic nuclei.

• For radiation therapy in the treatment of cancer, different cyclotrons are used.

• Cyclotrons can be used for nuclear transmutation (change of the nuclear structure).

Limitations of Cyclotron

• Neutral particles (e.g. neutron) do not interact with electric or magnetic fields. So, cyclotrons cannot be used to accelerate them.

• Since electrons have very small mass, their speed increases very rapidly and soon the resonance between the high voltage and the particle becomes lost. Hence, a cyclotron cannot accelerate electrons.

• Cyclotrons can accelerate particles to speeds much less than the speed of light (in the non-relativistic regime).

What Exactly is a Cyclotron?

A cyclotron is an apparatus for increasing the energy of charged particles or ions. E.O Lawrence and M.S Livingston devised it in 1934 to examine the nuclear structure. The cyclotron boosts the energy of charged particles by using both electric and magnetic fields. Cross fields are named such because both fields are perpendicular to each other.

Charged particles accelerate outwards from the centre of a cyclotron along a spiral route. A static magnetic field keeps these particles on a spiral route, while a rapidly shifting electric field accelerates them.

Cyclotron Principle of Operation

• A charged particle beam is accelerated in a cyclotron by applying a high-frequency alternating voltage between two hollow "D"-shaped sheet metal electrodes inside a vacuum chamber called the "dees."

• Dees are situated between the poles of an electromagnet, which produces a perpendicular static magnetic field B.

• The Lorentz force perpendicular to the particle's direction of motion causes the particle's path to bend in a circle due to the magnetic field.

• Between the dees, an alternating voltage of several thousand volts is supplied. By creating an oscillating electric field in the region between the dees, the voltage accelerates the particles.

• The voltage is set at a frequency that allows particles to complete one circuit in a single cycle. The frequency must be tuned to the particle's cyclotron frequency to achieve this situation.

Cyclotron Frequency Expression

f = \[\frac{qB}{2\pi m}\]

• The magnetic field strength is denoted by the letter B.

• q is the particle's electric charge.

• m is the relativistic mass of the charged particle.

Particle Energy Expression

The particles' energy is determined by the magnetic field's strength and the diameter of the dees.

The formula for calculating the centripetal force required to keep the particles in a curved path is:

Fc = \[\frac{m\nu ^{2}}{r}\]

Lorentz's force FB on the magnetic field B provides the force.

F\[_{B}\] = q\[\nu\] B

Equating the equations we get,

\[\frac{m\nu ^{2}}{r}\]

\[\nu\] = \[\frac{qBR}{m}\]

Hence, the output energy of the particle is given by the expression

E = \[\frac{q^{2}B^{2}R^{2}}{2m}\]

Cyclotron Applications

These were the best sources of high-energy beams for nuclear physics investigations for decades. These are, however, still used in this type of research.

Is there Anything that Cyclotron Can't do?

• Because electrons have such a little mass, a cyclotron cannot accelerate them.

• The use of a cyclotron to accelerate neutral particles is not possible.

• Due to the relativistic effect, it cannot accelerate positively charged particles with enormous masses.

 Did You Know?

• To accelerate particles with relativistic speed, synchrocyclotrons (frequency of the voltage is adjusted after each cycle), and isochronous cyclotrons (the magnetic field is adjusted) are used. These modifications are made to balance the increasing mass of the accelerating particle as its speed tends to the speed of light. 

• Ernest O. Lawrence invented the first-ever cyclotron at the University of California, Berkeley in 1932. It was a 69 cm diameter machine with a maximum energy of 4.8 MeV. He was awarded the Nobel Prize in 1939 for this invention. He also invented a synchrocyclotron in 1945. 

• The largest cyclotron is at TRIUMF (Canada’s particle accelerator centre), which has a diameter of 18 m and maximum energy of 520 MeV.

• The Superconducting Ring Cyclotron (SRC) can produce high-intensity beams of accelerated particles. At RIKEN, a large research institute in Japan, there is an SRC of 19 m diameter. It has six superconducting sectors.